An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions

Abstract

In this paper, we provide an approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions and discuss the application of this approach in some physical problems. Concretely, we construct the equations for these three quantities; this allows us to achieve them by directly solving equations. In order to construct the equations, we introduce shifted local one-loop effective actions, shifted local vacuum energies, and local spectral counting functions. We solve the equations of one-loop effective actions, vacuum energies, and spectral counting functions for free massive scalar fields in R n , scalar fields in three-dimensional hyperbolic space H 3 (the Euclidean Anti-de Sitter space AdS 3 ), in H 3 /Z (the geometry of the Euclidean BTZ black hole), and in S 1 , and the Higgs model in a (1 + 1)-dimensional finite interval. Moreover, in the above cases, we also calculate the spectra from the counting functions. Besides exact solutions, we give a general discussion on approximate solutions and construct the general series expansion for one-loop effective actions, vacuum energies, and spectral counting functions. In doing this, we encounter divergences. In order to remove the divergences, renormalization procedures are used. In this approach, these three physical quantities are regarded as spectral functions in the spectral problem.

A $$1+5$$ 1 + 5 -dimensional gravitational-wave solution: curvature singularity and spacetime singularity

“…After the extension of the spacetime by the transformation of coordinates (4.1), when In future studies, we can start from the solution given in the present paper to consider the influence of the singularity to quantum effects by calculating, e.g., the partition function and the one-loop effective action based on the heat-kernel method [39,40].…”

Heat kernel approach for confined quantum gas

“…Moreover, Kac hypothesized that the third term is proportional to the Euler-Poincaré characteristic number [8]. Recently, there are many researches on the calculation of heat kernels and the corresponding physical quantities [9,10]. The thermodynamic properties of gases in confined space have been widely investigated in recent years, including classical gases [11][12][13] and quantum gases [14][15][16].…”

Virial coefficients expressed by heat kernel coefficients

“…Since the first several heat kernel coefficients can be analytically expressed in terms of geometric invariants of the manifold [8,9,10], the asymptotic expansion of heat kernel, i.e., the heat kernel expansion, becomes a very important tool [8,11,12]. The heat kernel expansion has been applied in many fields of physics, such as quantum field theory [13,14,15], quantum gravity [16,17], and string theory [18]. In quantum statistical mechanics, the sum over the spectrum is a key component to calculate the grand partition function, so the heat kernel approach also plays an important role [11,19,20].…”